This is joint work with C. Xue. Let G be a split reductive group and X be a smooth proper curve over a finite field of characteristic p. The global Langlands correspondence is controlled by the cohomology sheaves of the stacks of global shtukas. I want to discuss a result concerning the local monodromy of these sheaves, namely, we show that taking the cohomology sheaves of global shtukas commutes with taking nearby cycles. I then want to put this result in context of a certain form of categorical local Langlands.